We consider solutions of the stationary Extended Fisher-Kolmogorov equation with general potential that are global minimizers of an associated variational problem. We present results that relate the global minimization property to a generalized concept of monotonicity of the solutions. This monotonicity can be described as the lack of intersections of the solution curve when projected onto the $(u,u')$--plane. Our method is based on applying a cut-and-paste argument in the space $H^2( {mathbb{R )$ to intersections in the $(u,u')$--plane. The statements and proofs are presented for the Extended Fisher-Kolmogorov equation, but the method can be directly extended to a wide class of fourth-order ordinary differential equations that derive from minimization problems.

None of the above, but in MSC2010 section 58Exx (msc 58E99), None of the above, but in MSC2010 section 34Cxx (msc 34C99), Homoclinic and heteroclinic solutions (msc 34C37), Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (msc 37J45), Action-minimizing orbits and measures (msc 37J50)
Modelling, Analysis and Simulation [MAS]

Peletier, M.A. (2000). Generalized monotonicity from global minimization in fourth-order ODEs. Modelling, Analysis and Simulation [MAS]. CWI.